Label Placement and graph drawing

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Label Placement and graph drawing

• Imo Lieberwerth

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Overview

• Introduction
• The Edge Label Placement problem
• The Multiple Label Placement problem

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Introduction

• Automated label placement originates from the
  Cartography
• Many years of research
• Three types
• Point labeling
• Line labeling
• Area labeling

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Point labeling

• Label features like
• cities
• nodes of a graph

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Line labeling

• Label features like
• Rivers
• streets

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Rules for line labeling

• The label must follow the shape of the line (but
  not to strict)
• At long lines the label must be repeated
• No extra white spaces between characters
• Vertical lines
• Left of the map First letter towards the bottom
• Right of the map First letter towards the top

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Rules according Imhof(1962)
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Area labeling

• Label features like
• Countries
• Oceans

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Rules for area labeling

• Stretch label over whole the area
• Areal labels preferably have the same shape
• Use horizontal or curved labels
• Evenly curved labels
• Repeat labels at suitable intervals

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Labeling quality

• Problem of assigning text labels to graphical
  features such that association of labels is clear
• The label assignment must be unambiguous
• Lot of work is done by Imhof and Yoeli

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Problematic cases
Problem association Thunder Bay and point
Problem readability name
Problem visibility highway 20
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Basic rules for labeling quality

• Labels should be easily read and easily and
  quickly located
• No overlap of a label with other labels or other
  graphical features are allowed
• Each label can be easily identified with exactly
  one graphical feature of the layout (the
  assignment is unambiguous)
• Each label must be placed in the best possible
  position

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COST-function

• A optimal assignment can produces labels that do
  not strictly follow the rules
• COST(i, j) is penalty for label j to edge i
• Penalty with respect to the ranking of label j
• Penalty for violating the basic rules

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Labeling space
Discrete labeling space
Continuous labeling space
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The ELP problem

• ELP Is the problem of placing text labels
  assigned to particular edges of a graph
• ELP problem has received little attention
• Complexity is solved in 1996 and is NP-Hard by
  Kakoulis and Tollis
• Most algorithms do not produce desired results
• Trapped in local optima
• Take exponential time
• The ELP-algorithm

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The optimal ELP problem
Question Find a labeling assignment that
minimizes the following function
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The ELP algorithm

• Assumptions
• All labels have the same size
• Each edge has only one label associated with it
• Any acceptable solution must guarantee
• Any label must be free of overlap (except his
  associated edge)
• Any label must be very close (touch) to its
  associated edge

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Main idea of the algorithm

• Divide the input drawing into horizontal strips
  of equal height
• Finite number of label positions
• Find the set of label positions for each edge
• Each position must be inside a strip
• Each position has to touch a edge but not
  intersect this edge (or other graphical feature)
• Each label position overlaps at most with one
  other label position

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Example
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Example
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Example
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Example
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Matching problem

• Transform ELP problem where each edge is matched
  to one of his label positions
• Group label positions together
• If two positions overlap they belong to the same
  group
• Else to a single member group
• Only one member of group in assignment
• Guaranties labels do not overlap each other

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Matching graph

• Define matching graph Gm(Ve ,Vg ,Em)
• Each node e inVe corresponds to an edge
• Each node r in Vg corresponds to a group of label
  positions
• Each edge(e, r) in Em connects a node e inVe to a
  node r in Vg if and only if e has a label
  positioning that belongs to group r
• Gm is a bipartite graph
• For each edge in Gm add the cost for assigning
  label

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Example
1,2
3,4
g1
g2
g3
g4
g5
g1
g3
g4
g5
g2
g6
g6
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ELP_Algorithm

• INPUT A drawing D of the graph G(V, E)
• OUTPUT A label assignment
• Split D into horizontal strips
• Find all label positions for each edge and
  construct the groups R
• Create the matching graph Gm for D
• Match edges to label positions , by finding a
  matching in Gm

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Finding a matching

• Theorem. Let A be the set of label positions for
  all edges of a drawing D of graph G(V, E). If
  every label position in A overlaps at most one
  other label position in A, then a maximum
  cardinality minimum weight matching of the
  corresponding matching graph Gm produces an
  optimal solution to the ELP problem with no
  overlaps.
• You can use any known algorithm to find a maximum
  cardinality minimum weight matching

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Fast Matching Heuristic

• INPUT Matching graph Gm
• OUTPUT A maximum cardinality matching for Gm
  with low weight
• IF the minimum weight incident edge a node in VE
  connects this node to a node in VG of degree
  1THEN Assign this edge as a matched edge
  update Gm
• IF a node in VE in has degree oneTHEN Assign
  its incident edge as a matched edge update Gm
• Repeat steps 1 and 2 until no new edge can be
  matched
• Delete all nodes of degree 0 from GmFOR each
  node e in VE DO Remove all but the two
  incidents edges of e with the least weight
• The remaining graph consists of simple cycles
  and/or paths. Find the only two maximum
  cardinality matchings for each component. Choose
  the matching of minimum weight.

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Step 1

• INPUT Matching graph Gm
• OUTPUT A maximum cardinality matching for Gm
  with low weight
• IF the minimum weight incident edge a node in VE
  connects this node to a node in VG of degree
  1THEN Assign this edge as a matched edge
  update Gm
• IF a node in VE in has degree oneTHEN Assign
  its incident edge as a matched edge update Gm
• Repeat steps 1 and 2 until no new edge can be
  matched
• Delete all nodes of degree 0 from GmFOR each
  node e in VE DO Remove all but the two
  incidents edges of e with the least weight
• The remaining graph consists of simple cycles
  and/or paths. Find the only two maximum
  cardinality matchings for each component. Choose
  the matching of minimum weight.

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Example step 1
Ve
Ve
Vg
Vg
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Step 2

• INPUT Matching graph Gm
• OUTPUT A maximum cardinality matching for Gm
  with low weight
• IF the minimum weight incident edge a node in VE
  connects this node to a node in VG of degree
  1THEN Assign this edge as a matched edge
  update Gm
• IF a node in VE in has degree oneTHEN Assign
  its incident edge as a matched edge update Gm
• Repeat steps 1 and 2 until no new edge can be
  matched
• Delete all nodes of degree 0 from GmFOR each
  node e in VE DO Remove all but the two
  incidents edges of e with the least weight
• The remaining graph consists of simple cycles
  and/or paths. Find the only two maximum
  cardinality matchings for each component. Choose
  the matching of minimum weight.

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Example step 2
Ve
Ve
Vg
Vg
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Fast Matching Heuristic

• INPUT Matching graph Gm
• OUTPUT A maximum cardinality matching for Gm
  with low weight
• IF the minimum weight incident edge a node in VE
  connects this node to a node in VG of degree
  1THEN Assign this edge as a matched edge
  update Gm
• IF a node in VE in has degree oneTHEN Assign
  its incident edge as a matched edge update Gm
• Repeat steps 1 and 2 until no new edge can be
  matched
• Delete all nodes of degree 0 from GmFOR each
  node e in VE DO Remove all but the two
  incidents edges of e with the least weight
• The remaining graph consists of simple cycles
  and/or paths. Find the only two maximum
  cardinality matchings for each component. Choose
  the matching of minimum weight.

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Fast Matching Heuristic

• INPUT Matching graph Gm
• OUTPUT A maximum cardinality matching for Gm
  with low weight
• IF the minimum weight incident edge a node in VE
  connects this node to a node in VG of degree
  1THEN Assign this edge as a matched edge
  update Gm
• IF a node in VE in has degree oneTHEN Assign
  its incident edge as a matched edge update Gm
• Repeat steps 1 and 2 until no new edge can be
  matched
• Delete all nodes of degree 0 from GmFOR each
  node e in VE DO Remove all but the two
  incidents edges of e with the least weight
• The remaining graph consists of simple cycles
  and/or paths. Find the only two maximum
  cardinality matchings for each component. Choose
  the matching of minimum weight.

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Example step 4
Ve
Ve
Vg
Vg
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Fast Matching Heuristic

• INPUT Matching graph Gm
• OUTPUT A maximum cardinality matching for Gm
  with low weight
• IF the minimum weight incident edge a node in VE
  connects this node to a node in VG of degree
  1THEN Assign this edge as a matched edge
  update Gm
• IF a node in VE in has degree oneTHEN Assign
  its incident edge as a matched edge update Gm
• Repeat steps 1 and 2 until no new edge can be
  matched
• Delete all nodes of degree 0 from GmFOR each
  node e in VE DO Remove all but the two
  incidents edges of e with the least weight
• The remaining graph consists of simple cycles
  and/or paths. Find the only two maximum
  cardinality matchings for each component. Choose
  the matching of minimum weight.

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Example step 5

• Traverse cycle (or path) and pick only the even
  or odd edges as part of the matching

Ve
Ve
Vg
Vg
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Remark

• Algorithm dont label horizontal edges

• Solution rotate the drawing

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Extension and results

• Post processing step
• Relaxing the restrictions by allowing labels to
  overlap their associated edges
• The algorithm works very well for hierarchical
  and other straight-line drawing
• Not suitable for orthogonal drawings

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Result algorithm
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The MLP problem

• Konstantinos G. Kakoulis and Ioannis G. Tollis.
  On the multiple label placement problem. In Proc.
  10th Canadian Conf. Computational Geometry
  (CCCG98), pages 6667, 1998.
• Each graphical feature is associated with more
  then one label
• Constraints example with single edge and source
  and target node
• Proximity Label Ls must be in close proximity
  with source node. Define maximum distance
• Partial Order The source node label must closer
  to the source node then the other label. Define a
  partial order
• Priority First important labels and then the
  other labels if there is a space

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Algorithm 1

• Algorithm for solving the MLP problem
• Iterative approach
• Main loop
• i-th iteration of the loop, assigns the i-th
  label for graphical feature
• This algorithm can take into account all three
  constraints
• This algorithm can use the ELP-algorithm

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Example
An orthogonal drawing A hierarchical drawing
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Algorithm 2

• Non-iterative fashion and extension of
  ELP-algorithm
• Sketch algorithm
• Find set of label positions for each graphical
  feature
• Label positions that overlap each other are
  grouped in clusters
• Each cluster is matched to at most one graphical
  feature using flow techniques

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Graphs

• First create bipartite graph matching graph Gm(Vf
  , Vc , Em)
• Each graphical feature is represented by a node
  in Vf
• Each cluster is represented by a node in Vc
• Each edge in Em connect
• Then transform Gm to a flow graph Gflow(s, t,Vf
  , Vc , Em)

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Flow graph

• The flow graph of algorithm 2

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Flow graph

• Assign capacities
• Each edge of original matching graph has capacity
  one
• Each edge (c, t) has capacity one
• Each edge (s, v) has capacity equal to the number
  of labels associated with v
• Include the cost of the label positioning for
  each edge
• Solve the maximum flow minimum cost problem for
  the flow graph Gflow

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Flow graph

• The flow graph of algorithm 2

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Example
Circular drawing with 3 labels per edge and node
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Experimental Results

• Both algorithms perform well and the same with
  respect to the success rate
• The flow method produces a slightly better
  quality of label assignments
• If labels have variable size the algorithm 1
  would be the best choice

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• Questions?